We have \(\nu _2(S(c 2^h,b 2^h+a))=\sigma _2(a)+f(b,c)=\sigma _2(a+1)-1+\nu _2(a+1)+f(b,c)\) 
if \(1\le a\le 2^{h-1}\)  and \(\nu _2((b 2^h+a+1) S(c 2^h,b 2^h+a+1))=\nu _2(a+1)+\sigma _2(a+1)+f(b,c)\)  if \(1\le a+1\le 2^{h-1}\) , i.e., \(1\le a \le 2^{h-1}-1\)  combined, by Theorem REF . Therefore, \(\nu _2(S(c 2^h+1,b 2^h+a+1))=\nu _2(S(c 2^h,b 2^h+a))=\sigma _2(a)+f(b,c)\)  which also generalizes [1]} that proved a conjecture by Amdeberhan et al.; cf. [2]}.
